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LaTeX source file
Appeared in Advances in Applied Mathematics v. 47(2011), 813819.
Written: Nov. 21, 2010.
In how many ways can a positivelywalking rook travel from the origin to
[n,n,...,n] in the ddimensional (hypercubic) lattice? Of course there is no closedform
formula, but for fixed d ( 2 ≤ d ≤ 12) we can get recurrences, and
also for fixed n and variable d.
If you could't care less about the number of ways a rook can walk, perhaps
you may be ineterested in finding out in how many ways you can pay
your d debts, to d different creditors, where you owe each n dollars.
It is the same answer! (why?)
Mathematica Output
Important: This article is accompanied by
Manuel Kauers' Mathematica output page
that has the Rook recurrences for dimensions up to 12 and the Queen recurrences up to 5.
Main Maple Output
The recurrences for dimensions 2 through 9, as well as the initial conditions, can also
be viewed in the file
ManuelRookRecurrences .

Using this file, as well as the Maple program
AsyRec
the
input file
yields the
output file
that gives the precise asymptotic formulas to order 10 for dimensions 2 through 9.

If you want to see the first 50 terms of the rookwalks enumeration sequences for dimension d (2 ≤ d ≤ 9)
using the recurrences found by Manuel Kauers' computer, and the initial d values,
the input file yields the
output file
Maple Package For Fixed n and Variable Dimension
The Maple package
RookWalks
handles this case.
Maple Output For Fixed n and Variable Dimension
If you want to see the first 150 terms of the sequence enumerating positive Rook walks from
the origin to [n,n,...,n] in the ddimensional hypercubic lattice for fixed n and variable d,
the guessed recurrences, and the implied asymptotic formula,
[whose leading term seems to be
e^{n1} (nd)!/n!^{d} ]
for n=1 (n!) to n=4, the
input file
yields the
output file
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